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Wednesday, March 24, 2010

the things money can't buy, part 2

It may be that some on the Mathematics Working Group, perhaps over-generalizing from their personal experience in school, underestimate the cost of bringing all children to mastery on, say, decimal long division with the traditional algorithm.

I have an idea.

How about the schools economize on curriculum materials by purchasing Primary Mathematics instead of Everyday Math, so we have enough money to teach everybody long division?

Boy, it really is like being a kid in a candy shop in terms of what statement is worthy of quoting as an example of their thoughtworld. Just when you think you have one, another more horrendous one comes along.

But this one was interesting:

The criteria that should be applied to decide which paper-and-pencil methods children should learn ought not to e the same today as they were 50 years ago. For example, today it may be bette to choose a paper-and-pencil algorithm that is more conceptually transparent and less efficient than the method that was optimal before the invention of computers and calculators. Or it may be a good idea to teach paper-and-pencil algorithms that promote students' proficiency with mental arithmetic.

With respect to the last sentence, the long division algorithm does a lot to promte number sense and proficiency with mental arithmetic.

OOH WAIT. Here's another good one:

Emphasis on a single,privileged method of solving a problem (such as a prescribed standard algorithm) leads to a rigidity of thought that impedes the development of number sense. By teaching a single algorithm to the exclusion of other methods, one runs the danger of raising a generation of non-thinkers who simply parrot processes they've been spoon fed.

Wow. A "single, privileged method of solving a problem". Sorta goes along with the privileged class don'tcha think?

Hey, Andy Isaacs! I know you're reading these comments. Let's hear from you. Who write this paragon of wisdom? You or Zal Usiskin?

As for the generation of non-thinkers the standard algorithms will produce, please note than in 1966, the number of students in the US who received bachelor's in math and engineering peaked. It would be interesting to see what the increase has been in the number of majors in communication arts and graphic design since the inception of the NCTM standards, EM, Investigations and other atrocities that seem to have a never ending supply of funding.

"By teaching a single algorithm to the exclusion of other methods, one runs the danger of raising a generation of non-thinkers who simply parrot processes they've been spoon fed."

So, if you can do long division, you're a non-thinker? Well, I suppose they're right. When I do long division, I don't want to waste time contemplating other ways to divide 652 by 12. I want to do the division problem as efficiently as possible, to be able to progress to more interesting things. It would be good if this took as little of my brain's processing capacity as possible.

To extend the metaphor, when I need to drive a nail into a 2 x 4, I don't want to forge an entirely new tool. I don't want to try to bash it in with a rock. I reach for my hammer, and I hammer the nail.

Some will remember this nugget from one of their papers called: "Algorithms in Everyday Math" from long ago.

"Reducing the emphasis on complicated paper-and-pencil computations does not mean that paper-and-pencil arithmetic should be eliminated from the school curriculum. Paper-and-pencil skillsare practical in certain situations, are not necessarily hard to acquire, and are widely expected asan outcome of elementary education. If taught properly, with understanding but without demandsfor “mastery” by all students by some fixed time, paper-and-pencil algorithms can reinforcestudents’ understanding of our number system and of the operations themselves. Exploringalgorithms can also build estimation and mental arithmetic skills and help students seemathematics as a meaningful and creative subject."

No talk of cost here. In fact it says that the skills are not necessarily hard to acquire.

But what happens when you get to dividing 3 by 2/5? Is this a paper-and-pencil skill? How about giving students the foundation for understanding how to divide something like X^2 by X/(x-1)?

Can someone dig up the old EM comment about how kids don't have to know how to invert and multiply because most adults don't need to do that?

Key words:

exploremeaningfulcreative

"...without demandsfor 'mastery' by all students by some fixed time, ..."

Trust the spiral.

There it is. low expectations. EM is not for the elite, like those privileged to be able to get to a real algebra class in 8th grade.

The problem is that Everyday Math doesn't expect mastery of ANY algorithm (even partial products or the lattice technique) at any point in time.

" ... but without demandsfor 'mastery' by all students by some fixed time"

This applies to everything they do. It defines their spiral. No mastery at any fixed time. Trust the spiral, then blame the student when they get to 7th grade.

"...algorithms can reinforcestudents’ understanding of our number system and of the operations themselves."

But how much do you understand if you can't do the problem using ANY algorithm, even one that you have discovered? There is not some sort of magical disconnect between understanding and algorithms, whichever one you use. You don't solve problems with understanding. You have to translate that into algorithm-like steps. So what EM does is to allow kids to spiral along not having to prove that they can translate their understandings into any particular algorithm to slove a problem.

They can't have it both ways. At some point, students have to be able to show their understanding by actually solving problems that do have just one answer.

Everyday Math reforces the elitism tag because all of the kids taking real algebra in 8th grade received help at home or from a paid tutor. The rest only get to say that they appreciate math as a creative endeavor. I doubt it.

If you can apply mathematical algorithms to solve problems, you have to have at least a little bit of understanding. This problem can be fixed. If you have understanding, but cannot translate that into steps to solve a problem, then you are nowhere. This cannot be fixed.

How about the schools economize on curriculum materials by purchasing Primary Mathematics instead of Everyday Math,

Wait, does Primary Math cost less than EM? What about Investigations?

That's just crazy, if that's the case. How can a school district justify spending more on a math program in this economy? There's got to be some leverage there, in terms of getting more parents organized to oppose the continuation of "discovery"-based math programs.

There's got to be a way to switch over without it costing that much money. If the materials cost less, that has to matter at some point down the road, doesn't it? Why should taxpayers put up with this?

Actually, when comparing Trailblazers to Singapore Math, the latter will come out more because the materials are "consumable". I.e., the texts and workbooks are meant to be written in. So you have to buy a new set of textbooks every year, whereas for Trailblazers, the same books are used over and over.

That said, EM's materials are conumable as well. If the choice were between EM and SM, I doubt if price would be a consideration based on what I'm hearing aobut school board decisions.

Singapore Math Textbooks can last for years if used gently in a classroom. We had our classroom set for 6 years, with periodic replacements needed. The workbooks are definitely consumable.

One main financial consideration between the EM/Trailblazers & SM series is that the EM series comes with training. Singapore Math materials do not come with training; schools need to add that cost into the overall purchase. Even with training, I can't see how it would cost a school district 100K unless they had close to 10,000 students.

(Catherine, tell Irvington I'll give them a discount ;)

The Home Instructors Guides are written by the U.S. Publisher, SingaporeMath.com. Jenny Hoerst, who has experience with homeschooling, is the main author of the books. They are frequently available on homeschooling sites and Ebay. (As are most of the SM materials, except the 3rd grade CWP!)

The Home Instructors Guides are written by the U.S. Publisher, SingaporeMath.com. Jenny Hoerst, who has experience with homeschooling, is the main author of the books. They are frequently available on homeschooling sites and Ebay. (As are most of the SM materials, except the 3rd grade CWP!)

Huh -- I was wondering about that, because when I was looking up the 2nd grade materials last week, it said "copyright Sonlight Curriculum" at the bottom of the HIG pages. But I see the first grade materials are copyright Singaporemath, so maybe SL did some and SM did others...

The HIG's for the new Standards edition were written by a different author than the HIG's for the U.S. edition. I've heard that the former are easier to use but I have no personal basis for comparison.

There are a couple of textbooks designed for teacher prep. courses that I've been considering getting for myself. They use the Singapore books to teach math to prospective teachers. They are by Parker & Baldridge and one deals with arithmetic (Elementary Math for Teachers) and the other deals with geometry (Elementary Geometry for Teachers). $28 each on Singaporemath.com